| The first major theorem in the study of complete two-dimensional minimally immersed surfaces in Euclidean space was Bernstein's theorem, stating that a complete minimal graph in 3-space must be a plane. Recent generalizations of this result include the work of Schoen/Fischer-Colbrie and DoCarmo/Peng showing that the only complete stable orientable minimal surface in 3-space is the plane, and the result of Micallef that complete stable orientable parabolic surfaces in 4-space are holomorphic curves.;Our research has centered on corresponding questions for complete nonorientable minimal surfaces. In 3-space we have used an idea of Ros/Lopez to show that there are no complete, nonorientable stable minimal surfaces of finite total curvature and low genus. Specifically, the orientable double cover of such a nonorientable surface cannot be of genus 0 or 1, or be hyperelliptic. This rules out genus 2 surfaces as well, since all genus 2 surfaces are hyperelliptic. If the surface is allowed to have finitely many branch points then these results continue to hold, except if the surface has odd genus ;In 3-space, we have also considered a seemingly distinct problem. We have been able to prove the stability of the classical Schwarz P and D surfaces with respect to all volume-preserving variations. Since stability of a minimal surface is a property of its Gauss map, we have been able to use this result to produce complete stable branched nonorientable minimal immersions of genus 3 and higher. Though one would certainly prefer to have unbranched examples, these surfaces are still of interest, since the stability results for orientable surfaces quoted above continue to hold in the presence of branch points.;In higher dimensions, we have considered the more restricted class of complete unoriented area-minimizing surfaces. Applying density results of Morgan, we have been able to show such surfaces have finite total Gaussian curvature. This has enabled us to prove that there do not exist non-trivial low genus examples. Specifically, there are no tori, projective planes or Klein bottles. However, there do exist unoriented area-minimizing spheres, but these are completely classifiable: they are either planes or complex hyperbolas. We have shown that these are the only complete unoriented area-minimizing holomorphic curves. |
